Question Detail
Question
The 2nd, 3rd and 9th terms of an AP are the three consecutive terms of a GP. Find the common ratio of the GP.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Let the first term of the AP be a and common difference d. Then the 2nd, 3rd, and 9th terms are a+d, a+2d, and a+8d respectively. Since these are consecutive terms of a GP, (a+2d)² = (a+d)(a+8d). Expanding: a² + 4ad + 4d² = a² + 9ad + 8d² => 0 = 5ad + 4d² => d(5a+4d)=0. If d ≠ 0, then 5a+4d=0 => a = -4d/5. The common ratio r = (a+2d)/(a+d) = (-4d/5+2d)/(-4d/5+d) = (6d/5)/(d/5) = 6. So r=6. But 6 is an option. However, if d=0, the AP is constant and the GP ratio is 1, not in options. With a = -4d/5, r = 6. So the common ratio is 6. Option C is 6.
Hint
For three terms in GP, the square of the middle term equals the product of the other two.